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Each bunch of protons circulating in the LHC at full power will have about 120 billion protons of 14 TeV each, i.e. $1.6 \times 10^{24}$ eV which is about 250000 Joules.

Additionally, the comment below says that there can be up to 2808 bunches in the ring at a given time. Note also that the per proton energy should be 7 TeV/proton, not 14 TeV.

The circumference of the LHC is 27 km. Dividing by the speed of light, you find that the particles take 90 μs to complete a trip around the ring. Combining this with the numbers given in the linked answer, we can answer your questions:

What is the total kinetic energy per second of the particles accelerated by the LHC?

The total kinetic energy per second (i.e. power) passing through a given slice of the beamline at full capacity is:

2808 * 125000 Joules / 90 μs $\approx$ 4 Terawatts

Related: how many protons per second can the LHC accelerate?

The rate of protons passing a given point is:

2808 * 60 billion protons / 90 μs $\approx$ 10^18 protons/second

However, this isn't really "many protons per second can the LHC accelerate", since the LHC operates in cycles of filling up with protons, accelerating, and dumping/colliding.

Note that the accelerator does not dissipate that much power (the loss of kinetic energy proceeds only through Bremstraulung and collision plus there are other inefficiencies in the system), but that the load due to a large accelerator is enough that the local power company in Illinois needed to be informed ahead of time before FNAL fired up the Tevatron.
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dmckee♦Mar 18 '12 at 18:56

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Note that the 4 Terawatts counts each particle about 11'000 times (the number of revolutions per second). You could not exploit this power: if you were to fully absorb the energy of the particles (e.g. at the beam dump), the machine would be empty after one revolution.
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Andre HolznerAug 18 '14 at 20:49